Boolean percolation (also called continuum percolation) is one of the models that describes percolation phenomena. Informally, Boolean percolation on a metric measure space can be explained as follows: rain falls across the space, forming many puddles. We assume that the shapes of these puddles are (metric) balls, and the density of raindrops per unit ball is parameterized by an intensity parameter. Furthermore, their radii are determined by some i.i.d. random variables. The Boolean percolation model is the union of such puddles.
In the $n$-dimensional Euclidean space, many researchers have studied Boolean percolation. For example, Gouéré (2008) showed that the nonexistence of an infinite cluster for some intensity is equivalent to the radius distribution having a finite $n$-th moment. In this seminar, I will discuss an extension of Gouéré's results to general metric measure spaces that possess a certain kind of global regularity, namely, Ahlfors regularity. Roughly speaking, this reveals that the number $n$ in Gouéré's result corresponds to the volume growth rate of the space. This talk is based on my recent result arXiv:2410.23859.